In this section we will deal with three cases in which the law of a mixture is determined not by a blanket rule, but by calculating the ratio between the components: a meduma mixture that fell into unconsecrated produce, leaven of Terumah, and drawn water that entered a mikvah.
The first case - a meduma mixture that fell into unconsecrated produce:
One unit of Terumah that fell into fewer than one hundred units of unconsecrated produce - the mixture becomes meduma, which is forbidden by rabbinic law. By Torah law, as explained, a majority of unconsecrated produce is enough; by rabbinic law a ratio of one hundred to one is required, and that ratio does not exist here. Let us assume, for the sake of discussion, that there were nine units of unconsecrated produce and one unit of Terumah fell in - ten units in all, so that the entire mixture is ten percent Terumah.
What is the law when one unit of this mixture falls into another quantity of unconsecrated produce? How much is required in the second batch of unconsecrated produce in order to permit the new mixture to be eaten?
Rabbi Eliezer: always adopts the most stringent possibility - perhaps what came out of the first mixture was entirely Terumah. Therefore we treat the meduma mixture, which is ten percent Terumah, as though it were pure Terumah. That is, if one liter of the meduma mixture fell into unconsecrated produce, one hundred liters of unconsecrated produce are needed to permit the second mixture.
The Sages: take only the precise ratio into account, on the assumption that the mixture is evenly blended. If the mixture was ten percent Terumah, then the liter that fell in contained only a tenth of a liter of Terumah. Therefore one hundred times that amount is enough - ten liters of unconsecrated produce in the second mixture - to permit it to be eaten.
This is the meaning of the wording of the Mishnah: a mixture of Terumah and unconsecrated produce does not prohibit the new mixture as though it were full-fledged Terumah, "ela lefi cheshbon" - rather, according to a calculation that tracks the true proportion of Terumah that actually fell in.
The Mishnah goes on to bring two additional cases that are likewise based on calculating ratios.
The second case - leaven of Terumah:
Let us begin with the background. There are many rules governing nullification: in a mixture of forbidden and permitted substances, a required ratio is set in order to give the mixture the status of being permitted, based on the assumption that the forbidden portion is too small to play any role in the mixture. Nullification means that the forbidden substance has, as it were, disappeared, and is not considered a reality that must be reckoned with. With Terumah, as noted, the ratio is one hundred to one: when there is one hundred times as much unconsecrated produce as the single unit of Terumah, the Terumah component is not significant enough and it is nullified.
There are, however, exceptions to the rules of nullification - things that are never nullified, and one of them is the case the Mishnah deals with here: leaven.
On Pesach the Torah forbade two things: chametz and leaven. Chametz is grain products that leavened in the normal way - one of the five species of grain that came into contact with water and, after eighteen minutes, became chametz. Leaven is something else: it is not a food and is not fit to be eaten, but it fills the role of what we call yeast today, since it contains yeast. Nowadays, someone who wants to bake bread adds pure yeast to the dough and it causes the dough to rise; in the time of the Mishnah there was no access to pure yeast, so they would separate off a little of the dough, leave it aside without baking it, and the next day mix the previous day's dough into the fresh dough. Yesterday's dough had fermented, and although it is not fit to be eaten, it serves as a catalyst for producing yeast and leavening the new dough. This is leaven.
Let us assume that the leaven itself is Terumah - for example, dough of challah that was separated and then used as the leavening agent placed into today's batch of dough. It would be a mistake to think that one hundred times as much unconsecrated dough against that unit of Terumah leaven is enough. For even if there were a thousand times as much unconsecrated dough against the leaven, one cannot say that the Terumah is nullified and gone, since the very fact that the bread rose proves that the leavening agent is at work here. The bread cries out that it is still present, and therefore it is not nullified. This category is called ma'amid - the component that brings about the leavening of the bread.
Another example to illustrate the point: rennet, an animal protein that serves as an enzyme added to milk in order to turn it into cheese. Even if there is a thousand times as much milk as the small amount of rennet, we cannot say that it is nullified, since the very fact that the milk has become cheese shows that the enzyme is at work inside it. The enzyme cries out "here I am," and for that reason it is not nullified.
Therefore, leaven of Terumah that fell into a batch of dough - in any amount whatsoever, even with a thousand times as much unconsecrated dough against it - if the dough rose, the dough is forbidden, and the Terumah is not nullified even at more than one in a hundred, since the Terumah component announces itself: "Here I am, I am the one that leavened this bread."
The case in the Mishnah is a dough that is itself a mixture of Terumah leaven and unconsecrated leaven. Such a dough, if left to sit, will rise like any dough, and it is certainly forbidden, as explained. But what is the law when part of that mixture falls into a new batch of unconsecrated dough? Do we treat it as pure leaven so that everything becomes forbidden, or do we take a different approach?
According to the Bartenura, the Mishnah is speaking about a calculation: if the Terumah component within the mixture that fell was in a quantity sufficient on its own to make the second dough rise, the dough is forbidden, since the Terumah is still significant. But if the Terumah component by itself could not have made the bread rise, the fact that a larger combination of Terumah and unconsecrated leaven fell in does not render the second dough forbidden.
The language of the Mishnah: "V'ein hamachmetz machmitz ela lefi cheshbon" - a leaven mixture composed of unconsecrated dough and Terumah that fell into another batch of unconsecrated dough is not considered to have leavened it and forbidden it in any amount whatsoever, but rather according to the calculation: how much Terumah is actually present in the mixture that fell, and whether that quantity alone would have raised the unconsecrated dough into bread.
The third case - drawn water in a mikvah:
The background to the matter: a valid mikvah requires forty se'ah of water, and this water must be rainwater (as we will call it for the sake of simplicity), and not water drawn by hand in a vessel. Once the pit contains forty se'ah of valid water, one may add cup after cup of drawn water up to a million gallons, and the mikvah remains valid.
But before the forty se'ah have been completed, the law is different. By Torah law a majority suffices: if the pit contained twenty-one se'ah of rainwater and nineteen se'ah of drawn water were added to it, the mikvah is valid, since the majority of its water is valid. Rabbinically, however, as long as there are not forty se'ah, adding drawn water disqualifies the mikvah - and already with a mere three lugin.
To illustrate the measures: a se'ah is about twelve liters - like a pack of six two-liter bottles. Forty se'ah is forty times twelve, about four hundred and eighty liters, and that is the amount of water required for a valid mikvah. A log is a five hundred milliliter bottle, and three lugin are a liter and a half. A liter and a half is one eighth of twelve liters, since there are twenty-four lugin in a se'ah, so three lugin are one eighth of a se'ah. Accordingly, pouring three lugin - one eighth of a se'ah, about a liter and a half of drawn water - into a mikvah that does not yet contain forty se'ah of rainwater disqualifies the entire mikvah by rabbinic law.
Nevertheless, the Sages built in a certain leniency, since the gap between the Torah's requirements for a mikvah and the rabbinic requirements is a large one. As long as the majority of the mikvah is filled with rainwater, one does not pour the drawn water directly into the mikvah - for if one did, even three lugin, a liter and a half, would disqualify it - but rather one pours the minority portion of the mikvah, such as the nineteen se'ah, onto the ground beside the mikvah, and the water flows in on its own from the ground. This is not direct pouring but indirect pouring, and it is called 'hamshachah' - from the word for drawing along, since the water is drawn along and enters by itself. In this manner the mikvah is valid, provided that the majority of its water is rainwater.
This is how the Bartenura explains the plain meaning of the Mishnah: "Ein hamayim she'uvin poslin es hamikvah ela lefi cheshbon" - water that was drawn but was not poured in directly, rather indirectly to the side of the mikvah, and entered it on its own through hamshachah, does not disqualify the mikvah, unless the calculation shows that this mikvah does not contain a majority of rainwater. As long as the majority of the mikvah is rainwater, the minority may be drawn water that arrived through hamshachah, and the mikvah is valid.
In summary: in all three cases in the Mishnah the law is determined not sweepingly but "according to the calculation": in a medumah mixture that fell into unconsecrated produce - according to the Sages we calculate the true proportion of the Terumah that fell, and not like Rabbi Eliezer, who views the entire mixture as Terumah; with leaven of Terumah - we examine whether the Terumah component alone was capable of making the second dough rise; and with drawn water that entered a mikvah through hamshachah - we examine whether a majority of rainwater remains in the mikvah.